To determine how many moles of [tex]\(\text{KMnO}_4\)[/tex] are consumed to form 0.886 moles of carbon dioxide ([tex]\(\text{CO}_2\)[/tex]), we can follow these steps:
1. Identify the coefficients from the balanced chemical equation:
[tex]\[
14 \, \text{KMnO}_4 + 4 \, \text{C}_3\text{H}_5(\text{OH})_3 \rightarrow 7 \, \text{K}_2\text{CO}_3 + 7 \, \text{Mn}_2\text{O}_3 + 5 \, \text{CO}_2 + 16 \, \text{H}_2\text{O}
\][/tex]
From the equation, the coefficient of [tex]\(\text{KMnO}_4\)[/tex] is 14 and the coefficient of [tex]\(\text{CO}_2\)[/tex] is 5.
2. Set up the mole ratio:
We use the coefficients from the balanced equation to set up a ratio. For every 14 moles of [tex]\(\text{KMnO}_4\)[/tex], 5 moles of [tex]\(\text{CO}_2\)[/tex] are produced:
[tex]\[
\frac{\text{moles of } \text{KMnO}_4}{\text{moles of } \text{CO}_2} = \frac{14}{5}
\][/tex]
3. Calculate the moles of [tex]\(\text{KMnO}_4\)[/tex] consumed:
Given that 0.886 moles of [tex]\(\text{CO}_2\)[/tex] are produced, we can use the ratio to find the moles of [tex]\(\text{KMnO}_4\)[/tex] consumed:
[tex]\[
\text{moles of } \text{KMnO}_4 = \left( \frac{14}{5} \right) \times 0.886
\][/tex]
4. Perform the multiplication:
[tex]\[
\text{moles of } \text{KMnO}_4 = 2.4808
\][/tex]
Thus, 2.4808 moles of [tex]\(\text{KMnO}_4\)[/tex] are consumed to form 0.886 moles of [tex]\(\text{CO}_2\)[/tex].
